Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian
Michael E. Filippakis Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2004, 3(4): 729-756 doi: 10.3934/cpaa.2004.3.729
We study nonlinear Dirichlet problems driven by the scalar $p$-Laplacian with a nonsmooth potential. First for the so-called "sublinear problem", under nonuniform nonresonance conditions, we establish the existence of at least one strictly positive solution. Then we prove two multiplicity results for positive solutions. The first concerns the "superlinear problem" and the second is for the sublinear problem. The method of proof is variational based on the nonsmooth critical point theory for locally Lipschitz functions. Our results complement the ones obtained by De Coster (Nonlin.Anal.23 (1995)).
keywords: multiple strictly positive solutions. generalized subdifferential Mountain Pass Theorem nonsmooth critical point theory nonsmooth Palais-Smale condition Locally Lipschitz function
Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case
Michael E. Filippakis Donal O'Regan Nikolaos S. Papageorgiou
Communications on Pure & Applied Analysis 2010, 9(6): 1507-1527 doi: 10.3934/cpaa.2010.9.1507
We consider a nonlinear elliptic equation of logistic type, driven by the $p$-Laplacian differential operator with a general superdiffusive reaction. We show that the equation exhibits a bifurcation phenomenon. Namely there is a critical value $\lambda_*$ of the parameter $\lambda>0$, such that, if $\lambda>\lambda_*$, the equation has two nontrivial positive smooth solutions, if $\lambda=\lambda_*,$ then there is one positive solution and finally if $\lambda\in (0,\lambda_*),$ then there is no positive solution.
keywords: comparison principle. upper-lower solutions $p$-Laplacian truncation techniques superdiffusive reaction linking sets mountain pass theorem

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